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NICTA Copyright 2011From imagination to impact “Optimal” policy Egalitarian standpoint –Protocol A: toss coin, if heads all item to agent1 otherwise all items to agent2 –Protocol B: toss coin, if heads then next item to agent1 otherwise next item to agent2 –Arguably B more egalitarian than A as each agent gets ½ items on average?

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NICTA Copyright 2011From imagination to impact Other properties This mechanism is Pareto efficient –We can't swap players between teams and have both captains remain happy –Supposing captains picked teams truthfully This mechanism is not envy free One agent might prefer items allocated to other agent

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NICTA Copyright 2011From imagination to impact Strategic play This mechanism is not strategy proof –Captain1 can get a better team by picking players out of order –No need for Captian1 to pick early on a player that he likes but Captain2 dislikes –And vice versa

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NICTA Copyright 2011From imagination to impact Strategic play What is equilibrium behaviour? –Nash equilibrium: no captain can do better by deviating from this strategy –Subgame perfect Nash equilibrium: at each move of this repeated game, play Nash equilibrium

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NICTA Copyright 2011From imagination to impact Strategic play With 2 agents –There is unique subgame perfect Nash equilibrium –It can be found in linear time Even though there is an exponential number of possible partitions to consider!

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NICTA Copyright 2011From imagination to impact Strategic play With 2 agents –There is unique subgame perfect Nash equilibrium –It can be found in linear time SPNE(P1,P2,policy) = allocate(rev(P1),rev(P2), rev(policy))

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NICTA Copyright 2011From imagination to impact Strategic play With k agents –There can be multiple subgame perfect Nash equilibrium –Deciding if utility of an agent is larger than some threshold T in any SPNE is PSPACE complete

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NICTA Copyright 2011From imagination to impact Disposal of items Other protocols possible E.g. captains pick a player for the other team Addresses an inefficiency of previous protocol One captain may pick player in early round that the other captain would happily give away

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NICTA Copyright 2011From imagination to impact Conclusions Many other possible protocols TwoByTwo: Agent1 picks a pair of items, Agent2 picks the one he prefers, Agent1 gets the other TakeThat: Agent1 picks an item, Agent2 can accept it (if they are under quota in #items) or lets Agent1 take it … Many open questions How to compute SPNE with disposal of items? How to deal with non-additive utilities?